Question

Find the elements \( a_{43} \) and \( a_{22} \) of the matrix:

\[ A = \begin{bmatrix} 0 & 1 & 0 \\ 2 & 0 & 2 \\ 0 & 3 & 2 \\ 4 & 0 & 4 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ -3 & 2 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} 0 & 1 & -1 & 2 & -2 \\ 3 & -3 & 4 & -4 & 0 \end{bmatrix} \]

Step 1: Dimensions

First matrix is \(4 \times 3\), second is \(3 \times 2\), third is \(2 \times 5\). Final product will be \(4 \times 5\). (Matrix multiplication rule)

Step 2: Compute \( D = AB \)

\[ D = \begin{bmatrix} -3 & 2 \\ 12 & 4 \\ -1 & 12 \\ 24 & 8 \end{bmatrix} \]

Step 3: Compute Final Matrix \( A = D \times C \)

We only need \( a_{22} \) and \( a_{43} \) —

Find \( a_{22} \)

Row 2 of \(D\): \( [12 \;\; 4] \) Column 2 of third matrix: \( [1 \;\; -3] \) \[ a_{22} = (12)(1) + (4)(-3) = 12 – 12 = 0 \] —

Find \( a_{43} \)

Row 4 of \(D\): \( [24 \;\; 8] \) Column 3 of third matrix: \( [-1 \;\; 4] \) \[ a_{43} = (24)(-1) + (8)(4) = -24 + 32 = 8 \] —

Final Answer

\[ a_{22} = 0, \quad a_{43} = 8 \]

Next Question / Full Exercise